The residual effect of a growth-decay mechanism and the distributions of covariance structures

The product of two independent or dependent scalar normal variables, sums of products, sample covariances, and general bilinear forms are considered. Their distributions are shown to belong to a class called generalized Laplacian. A growth-decay mechanism is also shown to produce such a generalized Laplacian. Sets of necessary and sufficient conditions are derived for bilinear forms to belong to this class. As a generalization, the distributions of rectangular matrices associated with multivariate normal random vectors are also discussed.     

On a Dualization of Graphical Gaussian Models: A Correction Note

Abstract.  We correct two proofs concerning Markov properties for graphs representing marginal independence relations.     

Multiple imputation of missing data with ante-dependence covariance structure

A controlled clinical trial was conducted to investigate the efficacy effect of a chemical compound in the treatment of Premenstrual Dysphoric Disorder (PMDD). The data from the trial showed a non-monotone pattern of missing data and an ante-dependence covariance structure. A new analytical method for imputing the missing data with the ante-dependence covariance is proposed. The PMDD data are analysed by the non-imputation method and two imputation methods: the proposed method and the MCMC method.     

Markov Properties for Acyclic Directed Mixed Graphs

We consider acyclic directed mixed graphs, in which directed edges ( x → y ) and bi-directed edges ( x ↔ y ) may occur. A simple extension of Pearl's d -separation criterion, called m -separation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the m -separation criterion for arbitrary distributions.     

Standard errors for EM estimation

The EM algorithm is a popular method for computing maximum likelihood estimates. One of its drawbacks is that it does not produce standard errors as a by-product. We consider obtaining standard errors by numerical differentiation. Two approaches are considered. The first differentiates the Fisher score vector to yield the Hessian of the log-likelihood. The second differentiates the EM operator and uses an identity that relates its derivative to the Hessian of the log-likelihood. The well-known SEM algorithm uses the second approach. We consider three additional algorithms: one that uses the first approach and two that use the second. We evaluate the complexity and precision of these three and the SEM in algorithm seven examples. The first is a single-parameter example used to give insight. The others are three examples in each of two areas of EM application: Poisson mixture models and the estimation of covariance from incomplete data. The examples show that there are algorithms that are much simpler and more accurate than the SEM algorithm. Hopefully their simplicity will increase the availability of standard error estimates in EM applications. It is shown that, as previously conjectured, a symmetry diagnostic can accurately estimate errors arising from numerical differentiation. Some issues related to the speed of the EM algorithm and algorithms that differentiate the EM operator are identified.     

Designing and integrating composite networks for monitoring multivariate gaussian pollution fields

Networks of ambient monitoring stations are used to monitor environmental pollution fields such as those for acid rain and air pollution. Such stations provide regular measurements of pollutant concentrations. The networks are established for a variety of purposes at various times so often several stations measuring different subsets of pollutant concentrations can be found in compact geographical regions. The problem of statistically combining these disparate information sources into a single 'network' then arises. Capitalizing on the efficiencies so achieved can then lead to the secondary problem of extending this network. The subject of this paper is a set of 31 air pollution monitoring stations in southern Ontario. Each of these regularly measures a particular subset of ionic sulphate, sulphite, nitrite and ozone. However, this subset varies from station to station. For example only two stations measure all four. Some measure just one. We describe a Bayesian framework for integrating the measurements of these stations to yield a spatial predictive distribution for unmonitored sites and unmeasured concentrations at existing stations. Furthermore we show how this network can be extended by using an entropy maximization criterion. The methods assume that the multivariate response field being measured has a joint Gaussian distribution conditional on its mean and covariance function. A conjugate prior is used for these parameters, some of its hyperparameters being fitted empirically.     

Evaluating the Use of Spatial Covariance Structures in the Analysis of Cardiovascular Imaging Data

This article investigates the choice of working covariance structures in the analysis of spatially correlated observations motivated by cardiac imaging data. Through Monte Carlo simulations, we found that the choice of covariance structure affects the efficiency of the estimator and power of the test. Choosing the popular unstructured working covariance structure results in an over-inflated Type I error possibly due to a sample size not large enough relative to the number of parameters being estimated. With regard to model fit indices, Bayesian Information Criterion outperforms Akaike Information Criterion in choosing the correct covariance structure used to generate data.     

Multivariate process dispersion monitoring without subgrouping

The memory-type adaptive and non-adaptive control charts are among the best control charts for detecting small-to-moderate changes in the process parameter(s). In this paper, we propose the Crosier CUSUM (CCUSUM), EWMA, adaptive CCUSUM (ACCUSUM) and adaptive EWMA (AEWMA) charts for efficiently monitoring the changes in the covariance matrix of a multivariate normal process without subgrouping. Using extensive Monte Carlo simulations, the length characteristics of these control charts are computed. It turns out that the ACCUSUM and AEWMA charts perform uniformly and substantially better than the CCUSUM and EWMA charts when detecting a range of shift sizes in the covariance matrix. Moreover, the AEWMA chart outperforms the ACCUSUM chart. A real dataset is used to explain the implementation of the proposed control charts.     

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.     

A Comparison of Correlation Structure Selection Penalties for Generalized Estimating Equations

Correlated data are commonly analyzed using models constructed using population-averaged generalized estimating equations (GEEs). The specification of a population-averaged GEE model includes selection of a structure describing the correlation of repeated measures. Accurate specification of this structure can improve efficiency, whereas the finite-sample estimation of nuisance correlation parameters can inflate the variances of regression parameter estimates. Therefore, correlation structure selection criteria should penalize, or account for, correlation parameter estimation. In this article, we compare recently proposed penalties in terms of their impacts on correlation structure selection and regression parameter estimation, and give practical considerations for data analysts. Supplementary materials for this article are available online.